# How can the graviton be both massless and self-interacting?

Gravity is non-linear, so if it is mediated by gravitons, gravitons must interact with each other. On the other hand, the effects of gravity moves with the speed of light, so if it is mediated by gravitons, gravitons must be massless.

How can gravitons interact with gravitons if they are not charged under gravity?

• IMO if you want to learn about gravity, there is really no substitute for General Relativity. The quantum "view" of gravitation seems unnatural and "forced" by comparison. Commented May 2, 2023 at 15:35
• Why would you say that gravitons must interact with each other? I'm not disagreeing, just asking. Commented May 2, 2023 at 15:43
• Gravity's very complicated, but we can invent a scalar field theory that's massless but self-interacting too, e.g. of Lagrangian density $\mathcal{L}=\partial_\mu\phi\partial^\mu\phi-\frac14\lambda\phi^4$ (it's massless because there's no $\phi^2$ term), with EOM $\square\phi=-\lambda\phi^3$.
– J.G.
Commented May 2, 2023 at 16:15
• Essentially same question for photons: physics.stackexchange.com/q/34352/2451 , physics.stackexchange.com/q/22876/2451 and links therein. Commented May 2, 2023 at 16:25
• Gravitons and photons have zero rest mass, but what gravitons couple to, is energy and momentum, and gravitons and photons carry energy and momentum, so they do interact with gravitons. Energy and momentum is the charge of gravity. Because all energy couple with gravitons, that is why the rest mass energy, the part that dominates at low speeds, appeared in Newtonian gravity. Commented May 3, 2023 at 2:27

Gravitons couple to energy-momentum tensor $$T_{\mu \nu}$$, not mass (which captures only the $$00$$ component of $$T_{\mu \nu}$$). That's why photons (which are also massless, like gravitons) interact with gravitons leading to bending of light (Gravitational lensing).
The idea is that a free graviton field has its own $$T_{\mu \nu}$$, which must be added to the matter source's $$T_{\mu \nu}$$ to ensure that the conservation equation/Bianchi identity holds. But note that this free graviton's $$T_{\mu \nu}$$ is quadratic in the graviton field (say $$h_{\mu \nu}$$). This $$O(h^2)$$ term in the equations of motion can only come from an $$O(h^3)$$ term in the Lagrangian. This means that we have to update our free graviton theory to one that has a cubic interaction. But then this $$O(h^3)$$ term in the Lagrangian produces an $$O(h^3)$$ energy-momentum tensor, which must again be added to the previously-obtained sum of energy-momentum tensors in the equations of motion. This $$O(h^3)$$ term in the equations of motion can only come from an $$O(h^4)$$ term in the Lagrangian. This leads to a quartic interaction for the graviton field. And so on, until we sum up the entire series to obtain non-linear Einstein's theory with $$n$$-point graviton-graviton interactions for every $$n$$ from $$n = 3$$ until infinity. This was discussed several decades ago, and is nicely summarized in Deser's 1969 paper.